Question

It takes 208.4 kJ of energy to remove 1 mole of electrons from an atom on the surface of rubidium metal. How much energy does it take to remove a single electron from an atom on the surface of solid rubidium? What is the maximum wavelength of light capable of doing this?

Short answer

The energy required to remove a single electron from an atom on the surface of solid rubidium is \(3.46 \times 10^{-19} \) J/electron. The maximum wavelength of light capable of doing this is 572 nm.

Step-by-step solution

Convert energy from moles to single electron

To find the energy required to remove one electron from an atom on the surface of solid rubidium, we can use the following conversion: Energy per single electron (J/electron) = Energy per mole (J/mole) / Avogadro's number (electrons/mole) Given: Energy per mole = 208.4 kJ/mole = 208400 J/mole (convert from kJ to J) Avogadro's number = 6.022 x 10^23 electrons/mole So, Energy per single electron = \( \frac{208400}{6.022 \times 10^{23}} \) J/electron

Calculate energy per single electron

We can now calculate the energy by inserting the given values: Energy per single electron = \( \frac{208400}{6.022 \times 10^{23}} = 3.46 \times 10^{-19} \) J/electron

Use Planck's equation to determine maximum wavelength

Next, we want to find the maximum wavelength of light capable of removing one electron from the atom. To do this, we'll use Planck's equation relating energy (E), wavelength (\( \lambda \)), and Planck's constant (h). \( E = \frac{hc}{\lambda} \) Where: E = 3.46 x 10^(-19) J/electron (from step 2) h = 6.626 x 10^(-34) Js (Planck's constant) c = 3.0 x 10^8 m/s (speed of light) We need to solve for λ: \( \lambda = \frac{hc}{E} \)

Calculate maximum wavelength

By inserting the values, we can find the maximum wavelength: \( \lambda = \frac{(6.626 \times 10^{-34})(3.0 \times 10^8)}{3.46 \times 10^{-19}} = 5.72 \times 10^{-7} \) m Now, let's convert this value to nanometers for a more convenient unit: \( \lambda = 5.72 \times 10^{-7} \times 10^9 = 572 \) nm So, the maximum wavelength of light capable of removing one electron from an atom on the surface of solid rubidium is 572 nm.

Key concepts

Planck's constant

Planck's constant is a fundamental constant in the physical sciences, symbolized by the letter \( h \). It is pivotal in the understanding of quantum mechanics. Planck's constant relates the energy of a photon to the frequency of its associated electromagnetic wave. The equation for this relationship is:
\[ E = h u \] Here, \( E \) represents the energy of the photon, \( h \) is Planck's constant, and \( u \) (nu) denotes the frequency of the photon.

• Value: Planck's constant is approximately \( 6.626 \times 10^{-34} \) Joule seconds (Js), a very tiny number reflecting the small energy packets involved in quantum events.
• Significance: It highlights the quantum nature of light and matter and plays a crucial role in calculations involving quantum phenomena.

To illustrate, in the photoelectric effect, Planck's constant helps calculate the energy required to eject electrons from metal surfaces when struck by light. This demonstration of quantized energy interactions between photons and electrons underscores Planck's constant as a cornerstone of quantum theory.

Avogadro's number

Avogadro's number is a key concept in chemistry that bridges the atomic scale with the macroscopic world. It defines the relationship between moles (a counting unit in chemistry) and individual molecules or atoms.
- **Definition:** Avogadro's number is approximately \( 6.022 \times 10^{23} \) entities per mole. This means one mole of any substance contains exactly \( 6.022 \times 10^{23} \) atoms, ions, or molecules.- **Usage:** It allows chemists to determine the actual amount of a substance they are dealing with when conducting experiments or manufacturing products.
In the context of the original exercise, Avogadro's number helps convert the energy needed to remove one mole of electrons from rubidium into the energy needed for a single electron. By dividing the energy per mole by Avogadro's number, one can find the energy for one electron, which is crucial for determining the behavior of single electrons and understanding microscopic phenomena.

Electron energy conversion

Electron energy conversion is an important concept when examining how energy is transferred or transformed in atomic-scale interactions. It involves understanding how much energy is necessary to move an electron from its position in an atom or molecule.
- **Photoelectric Effect:** In the situation described by the exercise, an external energy source, such as light, supplies energy to remove an electron from a metal like rubidium. This process, known as the photoelectric effect, demonstrates how light energy is converted into electrical energy. - **Calculations:** To calculate the energy required to remove one electron, divide the total energy for a mole by Avogadro's number. The resulting energy per single electron helps determine what kind of light (in terms of wavelength) can provide enough energy to dislodge the electron.
This conversion of light energy to electron kinetic energy is key in technologies like photovoltaics and photoelectron spectroscopy, where understanding and harnessing electron energy conversion allows for the design and creation of devices such as solar cells.